I am currently trying to evaluate the sum

\begin{equation} \sum_{k = 0}^N \binom{N}{k}e^{c k^2} \end{equation}

If it were $k^2 \to k$ this would readily give $\left(1+ e^{c}\right)^N$, but I was wondering if this slightly different sum would admit a simple expression as well, even in the large $N$ limit.

  • $\begingroup$ Here is a simple expression: $\approx e^{cN^2}$. Depending on your application this may be a good enough way to think about it. $\endgroup$ May 17 at 10:13


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